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Chapter 21: Problem 48

You have a wire of length \(L=1.00 \mathrm{m}\) from which to make the square coil of a dc motor. The current in the coil is \(I=1.7 \mathrm{A},\) and the magnetic field of the motor has a magnitude of \(B=0.34\) T. Find the maximum torque exerted on the coil when the wire is used to make a single-turn square coil and a two-turn square coil.

Short Answer

Expert verified
Single-turn torque is 0.036 Nm; two-turn torque is 0.018 Nm.

Step by step solution

01

Determine the Side Length of the Coil for Single-Turn

For a single-turn square coil, use the entire wire length to form the square. The perimeter of the square is equal to the length of the wire: \( L = 4s \). Thus, the side length \( s \) is given by: \[ s = \frac{L}{4} = \frac{1.00}{4} = 0.25 \text{ m} \].
02

Calculate the Area of the Coil for Single-Turn

The area \( A \) of a square is given by \( A = s^2 \). Substituting \( s = 0.25 \text{ m} \), we calculate the area: \[ A = (0.25)^2 = 0.0625 \text{ m}^2 \].
03

Calculate Maximum Torque for Single-Turn

The torque \( \tau \) on a coil is given by \( \tau = nIBA \sin \theta \), where \( n \) is the number of turns, \( I \) is the current, \( B \) is the magnetic field, and \( \theta \) is the angle between the field and coil normal. For maximum torque, \( \sin \theta = 1 \). Substituting \( n = 1 \), \( I = 1.7 \text{ A} \), \( B = 0.34 \text{ T} \), and \( A = 0.0625 \text{ m}^2 \):\[ \tau = 1 \times 1.7 \times 0.34 \times 0.0625 = 0.036125 \text{ Nm} \].
04

Determine Side Length for Two-Turn Coil

For a two-turn coil, the wire length shared by two squares gives \( 2L \) as the total perimeter \( 2L = 8s \). Solving for \( s \), \[ s = \frac{L}{8} = \frac{1.00}{8} = 0.125 \text{ m} \].
05

Calculate Area of the Two-Turn Coil

Using the same area formula for squares, the area \( A \) for a side \( s = 0.125 \text{ m} \) is: \[ A = (0.125)^2 = 0.015625 \text{ m}^2 \].
06

Calculate Maximum Torque for Two-Turn Coil

Using the same torque formula for \( n = 2 \) turns,\[ \tau = 2 \times 1.7 \times 0.34 \times 0.015625 = 0.0180625 \text{ Nm} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

dc motor
A DC motor is an essential electromechanical device that converts direct current (DC) electrical energy into mechanical motion. The core mechanism relies on the interaction between magnetism and electric current flowing through a coil. This coil is often part of a larger circuit, generating torque that turns the motor's rotor.

Here's how it works in simple terms:
  • When you pass current through a coil placed in a magnetic field, it experiences a force according to Lorenz's law. This is the foundation of the DC motor's operation.
  • By managing the direction of the current flow and using components like a commutator, DC motors produce continuous rotational motion, providing the torque needed to drive mechanical systems.
Understanding DC motors is fundamental when delving deeper into electric motor design, efficiency improvement, and robotics.
magnetic field
Magnetic fields are invisible forces that surround magnetic materials and electric currents, playing a critical role in the operation of motors. In our scenario with the DC motor, the magnetic field interacts with the current in the coil to produce motion. Let's explore some key points:
  • Magnetic fields are characterised by the magnetic flux density, commonly denoted as \( B \), and measured in teslas (T). In the problem, \( B = 0.34 \, \text{T} \).
  • This field exerts a force on the charged particles within the coil, generating motion depending on the arrangement and strength of both the field and the current.
  • The direction of the force is determined by the right-hand rule, helping us predict the behaviour of the coil within the magnetic field.
Thus, understanding magnetic fields is crucial for comprehending how electric and magnetic forces meld to drive mechanical systems.
current in coil
Current, denoted as \( I \), is the flow of electric charge through a conductor, such as the coil of wire in a motor. Here, the current in the coil is what allows the motor to do work by inducing a force in the magnetic field:
  • The problem specifies that the current is \( I = 1.7 \, \text{A} \), which is crucial for producing torque in a coil.
  • Electric current in the coil interacts with the magnetic field, and by using equations of electromagnetism, we can predict the resulting motion and torque.
  • Current direction matters: altering it changes the interaction dynamics, enabling the motor to maintain rotation in the desired direction.
A strong grasp of the role of current helps us harness electromagnetism to perform mechanical tasks efficiently.
square coil
A square coil is a loop of wire formed into a square shape, and it is pivotal in maximizing the exposure of wire to both magnetic fields and the forces generated within a DC motor.
  • In our exercise, a wire of length \( L = 1.00 \, \text{m} \) is used to form either a single-turn or a two-turn square coil.
  • The coil's effectiveness in generating torque relies heavily on its geometry. For a square coil: the perimeter \( L = 4s \), and each side length \( s \) can be calculated easily.
  • A single-loop coil utilizes the entire wire length to form a larger square, whereas a double-loop divides the wire, forming smaller squares. The side length directly affects the coil鈥檚 surface area, which in turn influences the torque generated.
Understanding how coil structure and turns apply to torque calculations helps optimize motor design and performance.

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Most popular questions from this chapter

In New England, the horizontal component of the earth's magnetic field has a magnitude of \(1.6 \times 10^{-5}\) T. An electron is shot vertically straight up from the ground with a speed of \(2.1 \times 10^{6} \mathrm{m} / \mathrm{s} .\) What is the magnitude of the acceleration caused by the magnetic force? Ignore the gravitational force acting on the electron.

(a) A proton, traveling with a velocity of \(4.5 \times 10^{6} \mathrm{m} / \mathrm{s}\) due east, experiences a magnetic force that has a maximum magnitude of \(8.0 \times 10^{-14} \mathrm{N}\) and a direction of due south. What are the magnitude and direction of the magnetic field causing the force? (b) Repeat part (a) assuming the proton is replaced by an electron.

A square coil and a rectangular coil are each made from the same length of wire. Each contains a single turn. The long sides of the rectangle are twice as long as the short sides. Find the ratio \(\tau_{\text {square }} / \tau_{\text {rectangle }}\) of the maximum torques that these coils experience in the same magnetic field when they contain the same current.

A charge of \(-8.3 \mu \mathrm{C}\) is traveling at a speed of \(7.4 \times 10^{6} \mathrm{m} / \mathrm{s}\) in a region of space where there is a magnetic field. The angle between the velocity of the charge and the field is \(52^{\circ} .\) A force of magnitude \(5.4 \times 10^{-3} \mathrm{N}\) acts on the charge. What is the magnitude of the magnetic field?

An electron is moving through a magnetic field whose magnitude is \(8.70 \times 10^{-4} \mathrm{T}\). The electron experiences only a magnetic force and has an acceleration of magnitude \(3.50 \times 10^{14} \mathrm{m} / \mathrm{s}^{2} .\) At a certain instant, it has a speed of \(6.80 \times 10^{6} \mathrm{m} / \mathrm{s}\). Determine the angle \(\theta\) (less than \(90^{\circ}\) ) between the electron's velocity and the magnetic field.
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